Determine the value of the following complex number power. Your answer will be plotted in orange. $ ({\cos(\frac{5}{3}\pi) + i \sin(\frac{5}{3}\pi)}) ^ {15} $
Solution: Let's express our complex number in Euler form first. $ {\cos(\frac{5}{3}\pi) + i \sin(\frac{5}{3}\pi)} = { e^{5\pi i / 3}} $ Since $(a ^ b) ^ c = a ^ {b \cdot c}$ $ ({ e^{5\pi i / 3}}) ^ {15} = e ^ {15 \cdot (5\pi i / 3)} $ The angle of the result is $15 \cdot \frac{5}{3}\pi$ , which is $25\pi$ $25\pi$ is more than $2 \pi$ . It is a common practice to keep complex number angles between $0$ and $2 \pi$ , because $e^{2 \pi i} = (e^{\pi i}) ^ 2 = (-1) ^ 2 = 1$ . We will now subtract the nearest multiple of $2 \pi$ from the angle. $ 25\pi - 24\pi = \pi $ Our result is $ e^{\pi i}$. Converting this back from Euler form, we get $\cos(\pi) + i \sin(\pi)$.